rank of a matrix based on rank of its submatrices

Question

Let $G\in\mathbb{C}^{m\times n}$, and assume ${\rm rank~}G=m$. Define \begin{equation} H\triangleq\left[\begin{array}{cc}{\rm Re~}G&{\rm Im~}G\\-{\rm Im~}G&{\rm Re~}G\end{array}\right]\in\mathbb{R}^{2m\times 2n}. \end{equation} Show that ${\rm rank~}H=2m$.

Answer 1

Hint: Show that for $x,y \in \Bbb R^n$, $$ G(x - iy) = 0 \iff H \pmatrix{x\\y} = 0 $$